1.
Understanding numbers is fundamental and
the basis for all of Mathematics.
2. Every student
at every level of education must have a firm
understanding of
number.
Various
systems of writing numbers can be and should be addressed
at later stages of the student's development, but these
systems are best understood once the base 10 system is firmly
understood.
Why base 10?
well, we have 10 fingers; fundamentally, counting is done with
our fingers at the earliest level. So, it is normal to
count using our fingers using one after the other like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Now, we've run out of
fingers, I suppose we could then use our toes, but then at 20
we'd run out both fingers and toes... time to bring your
friends into the picture! No, it's probably best to
group by tens and write down a mark for each '10' we
pass by; then when we're done counting the number of 10s we
have is the number of marks we wrote down and the excess would
be the number of fingers we used past the last 10.
Now using both we'd have the number of objects we were
counting.
(Zero, '0', is
problematic; without resorting to meta mathematics, that is,
algebra theory, explaining zero can be a challenge;
HOWEVER, zero DOES NOT mean "NOTHING." It
is the identity of the binary operation addition in
the group of objects we normally call numbers, and further,
acts as a placeholder for every base-n number system we choose
to use to represent numbers. Teaching zero as
a number does not work well with our fingers, it is probably
best to introduce it as a placeholder when the time comes to
actually write the numbers we speak.)
Numbers as
quantity; approach numbers this way at first.
Cardinality refers to quantity, we are therefore talking about
cardinal numbers (ordinal numbers will have to wait!)
Numbers mean,
then, "how many." And this is the
approach to take no matter the student's grade level and
ability. Use a hands on approach counting similar
objects, always count similar objects (doing so reinforces
later algebra concepts.) Count books, count
pencils, count pens, count marbles, count dolls, count
pennies, count quarters, count dollars, count whatever, but do
not count "things." Again, this is to
reinforce algebra skills later on.
Move the
objects from a larger pile to smaller piles in groups of ten,
starting with one when counting. 1,2,3,4,5,6,7,8,9,10...
one group, then start on the next group
11,12,13,14,15,16,17,18,19,20 ... group 2, etc. Making
smaller groups has the added benefit of not losing track of
where you were counting if you get interrupted, i.e., count
the groups 10, 20, 30, etc, then move into the partial,
remaining group and continue 61, 62, 63, 64 , for
example.